Real-rooted atlas

A207824 → basis A013609 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−5, −3)coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(4 + x + 4\,n + n x\right)\,P_{n-1}(x) + \left(15 + 8\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18210
26332499
3496380968980
4390540001524256169701
530744394302009650886403296030
62420473729282381048064015280153664950599
719056323428088263059211162402828804281635841289409960
8150030093086304027662808141104644480240906752114240819225693149001
91181184402734822502803979521670930886377996816171680272384029388818432512922080050
109299445112393231840276248530818834301448399456802560266245401984077905927349764096010249127651499
11732143764820732261492266059481922042531868010422645760371193267294148812817007052821442560179712090112204890354434940
1257641556673178107735392251549623212214731015680123391857760502847262721490175052832357498885109465605722112043141121966084096894416697901
134538110157361519412734990234200650828822005220531841406174247680645349593024218841968640555194296321053762969614778649601488650241019904042598481928853812544070
1435728465692151288481617600021523327913072220734584263681552704623004879248605132803026533184640878603624448194828950272328409579524142336000379125760238018569175041638487643708742799

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(8 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 2\,x\)
Technical details
Node id
n-ee28494fe17a8fcd · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-fc39265b029c83a0)
Fingerprint
f6-a644861d2626c5b5

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         8,         2;
  2:        63,        32,         4;
  3:       496,       380,        96,         8;
  4:      3905,      4000,      1524,       256,       16;
  5:     30744,     39430,     20096,      5088,      640,       32;
  6:    242047,    372928,    238104,     80640,    15280,     1536,      64;
  7:   1905632,   3428088,   2630592,   1116240,   282880,    42816,    3584,    128;
  8:  15003009,  30863040,  27662808,  14110464,  4480240,   906752,  114240,   8192,   256;
  9: 118118440, 273482250, 280397952, 167093088, 63779968, 16171680, 2723840, 293888, 18432, 512;
%F Formula
The row polynomials satisfy n*P(n,x) = (4*n+4+(n+1)*x)*P(n-1,x) + (15+8*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 4 + (n + 1)*x)*P[n - 1, x] + (15 + 8*x + x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 4 + (n + 1)*x)*v[i - 1] + (15 + 8*x + x^2)*deriv(v[i - 1], x))/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023